Metamath Proof Explorer


Theorem imbi2d

Description: Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 11-May-1993)

Ref Expression
Hypothesis imbid.1 ( 𝜑 → ( 𝜓𝜒 ) )
Assertion imbi2d ( 𝜑 → ( ( 𝜃𝜓 ) ↔ ( 𝜃𝜒 ) ) )

Proof

Step Hyp Ref Expression
1 imbid.1 ( 𝜑 → ( 𝜓𝜒 ) )
2 1 a1d ( 𝜑 → ( 𝜃 → ( 𝜓𝜒 ) ) )
3 2 pm5.74d ( 𝜑 → ( ( 𝜃𝜓 ) ↔ ( 𝜃𝜒 ) ) )